201 Quiz 1 FRI solution - 110.201 Linear Algebra 1st Quiz Solution Problem 1 Given the following system of equations 3x 2y 5z = 1 4x y z = 0 x-z =2 find

# Linear Algebra with Applications (3rd Edition)

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110.201 Linear Algebra 1st Quiz Solution February 15, 2005 Problem 1 Given the following system of equations: 3 x + 2 y - 5 z = 1 4 x - y + z = 0 x - z = 2 find all solutions using Gauss-Jordan elimination procedure. Is this an example of consistent system? Why? Solution: We use Gauss-Jordan: 3 2 - 5 4 - 1 1 1 0 - 1 1 0 2 We rewrite the matrix as : 1 0 - 1 3 2 - 5 4 - 1 1 2 1 0 - 3 × (I) - 4 × (I) then: 1 0 - 1 0 2 - 2 0 - 1 5 2 - 5 - 8 ÷ 2 then: 1
1 0 - 1 0 1 - 1 0 - 1 5 2 - 5 2 - 8 +(II) then: 1 0 - 1 0 1 - 1 0 0 4 2 - 5 2 - 21 2 ÷ 4 then: 1 0 - 1 0 1 - 1 0 0 1 2 - 5 2 - 21 8 +(III) +(III) then we have: 1 0 0 0 1 0 0 0 1 - 5 8 - 41 8 - 21 8 Therefore, we have the answer : x = - 5 8 y = - 41 8 z = - 21 8 Problem 2 Find the rank of the following matrix - 1 3 8 - 2 1 - 1 3 9 - 1 3 1 - 3 - 9 1 - 3 0 0 0 0 2 Solution: We find reduced row-echelon form of this matrix: - 1 3 8 - 2 1 - 1 3 9 - 1 3 1 - 3 - 9 1 - 3 0 0 0 0 2 × ( - 1) +(II) ÷ 2 2
then: 1 - 3 - 8 2 - 1 - 1 3 9 - 1 3 0 0 0 0 0 0 0 0 0 1 +(I) then: 1 - 3 - 8